Coherent structures in wall turbulence Smallest scale of the flow: kolmogorov scale (in the near atmosphere about 1mm) Largest scale of the flow: several times the boundary layer height (in the atmosphere may go up to O(1-10 Km ) There are 6-7 orders of magnitude ! However IF, we understand how turbulent structures behave and IF these structures truly play a major role (statistically) on momentum, scalar and energy fluxes, mixing, etc. ... Then we could propose low dimensional models, smart closures, control systems Short term goal: understand and control near wall processes (relevant for drag, lift, resuspension, etc) Long term goal: shift turbulent closure to larger scales, in order to solve large domain accurately (atmosphere, rivers, oceans) Flow visualization and sketches Kline 1967 (near wall streaks) (log and outer layer) (1981) Acarlar and Smith, 1987, downstream of a fixed hemisphere downstream of a low momentum fluid ejection Laminar flow upstream Kline 1967 Flow visualization (hydrogen bubbles, flow markers) Robinson 1991 Hairpin vortex detection: track of a strongly 3D structure on the 2D streamwise - wall normal laser sheet: Adrian et al 2000 The Biot-Savart law is used to calculate the velocity induced by vortex lines. For a vortex line of infinite length, the induced velocity at a point is given by: V = 2 πΓ /d where Γ is the strength of the vortex d is the shortest distance from a point P to the vortex line For a arch-like vortex line, there is a combined induction towards its center (ejection of low momentum fluid u’v’ Q2 event Q2 event vortex Shear layer A brief summary . . . • Single hairpin vortices can explain the observed features of low and high speed streaks, bursting phenomena and lift up of structures (viscous & buffer layer) • What is still missing so far is the outer layer, • Structures were observed to form bulges with ramp-like features. Numerical Simulation (Zhou, Adrian et al. 1996, 1999) isovorticity surface Self sustaining mechanism (see also Waleffe 1990) and vortex alignment Limitation : low Re with initial perturbation Experimental evidence of hairpin packets in smooth wall turbulence (Adrian, Meinhart, Tomkins JFM, 2000) Instantaneous flow fields: U-Uc (convection velocity) Vortex marker: swirling strength Q4 Q4 Ramp packet Q2 Detection of zones of uniform momentum associated to the streamwise alignment of hairpin vortex: mutual induction of Q2 event Vortex identification Okubo-Weiss parameter Q S 2 z2 S 2 Sn2 Ss2 where : Sn ux wz Ss wx uz From the local velocity gradient tensor u x u w x Swirling Strength analysis u z w z See also Chong & Perry, 1990 Jeong and Hussain 1995 Imaginary eigenvalues c cr ici We select the region where ci 0 Numerical simulation Adrian, PoF 2008 multimedia appendix Flow visualization, Statistical Signature 1)Relevance 2)Physical mechanisms 3)Connection with quadrant analysis (Lu & Willmart, 1973, Wallace 1972, Nezu & Nakagava 1977) 4)Vortex identification in 2D and 3D 5)Zones of uniform momentum 6)Consistency with observed resuspension events (strong correlation between c’w’ and u’w’ events) Besides instantaneous realizations… Is it possible to obtain some quantitative information about turbulent structures ? 2 point correlation 2 point correlation tensor Rij rx , y, y ' ui x, y u *j x rx , y ' correlation coefficient (normalized) Rij rx , y, y ' ρ ij for i, j u, v σ i y σ j y' See also Proper Orthogonal Decomposition (Holmes & Lumley ) Linear stochastic estimate Estimate of the flow field Statistically conditioned To the realization of a known event : 1) II quadrant (u < 0, v > 0) 2) IV quadrant (u > 0, v < 0) 3) Vortex identified by the swirling strength complex part of the eigenvalue of the local velocity gradient tensor. Comparison A B center (reduction of the streamwise lengthscale: lost of coherence within the structures of the packets) (see also Krogstad e Antonia 1994 rough wall) A B Two point correlation streamwise velocity fluctuation Comparison A B center A B Linear Stochastic Estimate: Question: What is the average flow field statistically conditioned to the realization of a vortex with a spanwise axe (identified as the signature of the hairpin vortex On the laser sheet)? The best (linear) estimate is given by u j x' λ x u j x' λx L jλx λ x λ x λ x con x ( x, y ) λx u j x' λx, y' u x rx , y Adrian, Moin & Moser, 1987 Adrian 1988, Christensen 2000 Note: Information about conditioned probabilistic variables are obtained from unconditioned statistical moments Linear stochastic Estimate : E known event assumed at a fixed y’ See Christensen 2000 Flow field obtained from a statistical analysis (conditioned to the realization of a E event) E Kim and Adrian 1999 Spanwise alignment of hairpin structures leading to long coherent regions of uniform momentum Kim & Adrian 1999 VLSM Contribution : turbulent kinetic energy and Reynolds stresses Guala et al. 06 Net force exerted by Reynold stress in the mean momentum equation • Pipe flow • Turbulent Boundary layers • channel flow Pipe : Guala et al, 06 channel: Turb. B. layer: Balakumar, (2007) A brief summary . . . • Large scale motion participate significantly to the Reynolds stress, thus contribute not only to TKE but also to TKE production. • In terms of momentum balance, close to the wall, VLSM push the flow forward, while smaller scales slow down the flow. • Such features are observed for turbulent pipe, channel and boundary layers flows Marusic & Hutchins 2008 Atmospheric Surface Layer Reτ=O(106) Hutchins & Marusic 2007 High Re Large scale influence on the near Wall turbulence intensity: Amplitude modulation Low Re High Re Low Re Note that in different research field some type of very large scale motions are addressed with different names e.g. streamwise rolls (atmospheric science) or secondary current (river hydraulics) VLSM : A visual inspection PIPE FLOW Lekakis 88, ATMOSPHERIC SURFACE LAYER (ASL) Metzger et al. 07; Guala, Metzger, McKeon 08 Chacin & Cantwell 2000 (Turb. Boundary Layer) Soria 94 Chong & Perry 90 A different view Luthi 2005 PTV isotropic turbulence Coherent structures vs vortices Open questions: 1) How spanwise mean vorticity relates to streamwise fluctuating vorticity ? 2) How vortex stretching is affected by a non zero mean strain ( and perhaps also mean vorticity) ? 3) Do they both scale with Kolmogorov (core) and the integral lengthscale ? 4) Are they more or less stable as compared to worms in isotropic 3D turbulence ? Other Questions 1) How roughness in general can perturb self organization, how about complex terrain ? 2) What are the relevant scales for coherent structures (inner, outer)? 3) Can we really define a coherent structure 4) Can we describe coherent structures evolution in unambiguous quantitative (not handwavy) terms ? 5) How VLSM rtelate to hairpin packets (is it Reynolds dependent)? 6) why near wall peak can be affected by outer layer structures? 7) which terms of which equation are responsible? 8) can we go beyond geometrical characteristics (exp) and vorticity contour (num) ?
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